I would totally not recommend to disable it. You should spend more time to figure the problem out. In my BuddyPress test installations (local and live), it sends the emails just fine, so there is a problem, maybe in your server, template, I don’t know, somewhere.

But, if you really wanna disable this, after a quick search here and in Google, I found some possible solutions.

But since it is holding us moving ahead since people can’t use site, so I think at least an option to disable sending the activation email (i.e. bypassing activation) will be good while people are trying to resolve this issue.

Also it will help let people use the site. Currently people just give up after seeing the “Invalid Activation Key” message and don’t return back to the site.